Solve -3/2+3i/2/sqrt{2}

$$- \frac { \frac { 3 } { 2 } + \frac { 3 i } { 2 } } { \sqrt { 2 } }$$

Evaluate

$$\sqrt{2}\left(-\frac{3}{4}-\frac{3}{4}i\right)\approx -1.060660172-1.060660172i$$

Solution Steps

Divide 3i by 2 to get \frac{3}{2}i.

$$-\frac{\frac{3}{2}+\frac{3}{2}i}{\sqrt{2}}$$

Rationalize the denominator of \frac{\frac{3}{2}+\frac{3}{2}i}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$-\frac{\left(\frac{3}{2}+\frac{3}{2}i\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$-\frac{\left(\frac{3}{2}+\frac{3}{2}i\right)\sqrt{2}}{2}$$

Divide \left(\frac{3}{2}+\frac{3}{2}i\right)\sqrt{2} by 2 to get \left(\frac{3}{4}+\frac{3}{4}i\right)\sqrt{2}.

$$-\left(\frac{3}{4}+\frac{3}{4}i\right)\sqrt{2}$$

Multiply -1 and \frac{3}{4}+\frac{3}{4}i to get -\frac{3}{4}-\frac{3}{4}i.

$$\left(-\frac{3}{4}-\frac{3}{4}i\right)\sqrt{2}$$

Real Part

$$-\frac{3 \sqrt{2}}{4} = -1.0606601717798214$$